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Read this text. Pay close attention to the "How To" sections which give summaries of how to do mean, median, and mode calculations. After you read the text, complete the practice problems and check your answers.

One application of decimals that arises often is finding the average of a set of numbers. What do you think of when you hear the word average? Is it your grade point average, the average rent for an apartment in your city, the batting average of a player on your favorite baseball team? The average is a typical value in a set of numerical data. Calculating an average sometimes involves working with decimal numbers. In this section, we will look at three different ways to calculate an average.

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Suppose Ethan's first three test scores were , , and . To find the mean score, he would add them and divide by .

His mean test score is points.

The **mean **of a set of numbers is the arithmetic average of the numbers.

**Calculate the mean of a set of numbers.**

Step 1. Write the formula for the mean

Step 3. Count the number, , of values in the set. Write this number in the denominator.

Step 4. Simplify the fraction.

Step 5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.

Did you notice that in the last example, while all the numbers were whole numbers, the mean was **59.5**, a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.

Ann | Bianca | Dora | Eve | Francine |
---|---|---|---|---|

59 | 60 | 65 | 68 | 70 |

Table 5.6

Dora is in the middle of the group. Her height, **65''**, is the median of the girls' heights. Half of the heights are less than or equal to Dora's height, and half are greater than or equal. The median is the middle value.

- Half the data values are less than or equal to the median.
- Half the data values are greater than or equal to the median.

What if Carmen, the pianist, joins the singing group on stage? Carmen is **62** inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:

There is no single middle value. The heights of the six girls can be divided into two equal parts.

Notice that when the number of girls was **5**, the median was the third height, but when the number of girls was** 6**, the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.

Step 1. List the numbers from smallest to largest.

Step 2. Count how many numbers are in the set. Call this .

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the **mode**. The mode of a set of numbers is the number that occurs the most. The **frequency**, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

Suppose Jolene kept track of the number of miles she ran since the start of the month, as shown in Figure 5.7.

Figure 5.7

If we list the numbers in order it is easier to identify the one with the highest frequency.

Jolene ran **8** miles three times, and every other distance is listed only once. So the mode of the data is **8 **miles.

Step 1. List the data values in numerical order.

Step 2. Count the number of times each value appears.

Step 3. The mode is the value with the highest frequency.

Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.

Source: Rice University, https://openstax.org/books/prealgebra/pages/5-5-averages-and-probability

This work is licensed under a Creative Commons Attribution 4.0 License.